Application of K\"ahler manifold to signal processing and Bayesian inference

TL;DR

This paper explores how K"ahler manifolds can simplify the information geometric analysis of linear systems and Bayesian inference, leading to easier calculations of geometric quantities and improved prior construction.

Contribution

It identifies conditions for the information geometry of linear systems to be K"ahler and demonstrates the benefits of this structure in Bayesian inference.

Findings

K"ahler structure simplifies geometric calculations

Conditions for linear systems to have K"ahler geometry identified

Improved Bayesian priors using K"ahler geometry

Abstract

We review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the K\"ahler manifold case. We find conditions for the information geometry of linear systems to be K\"ahler, and the relation of the K\"ahler potential to information geometric quantities such as $\alpha $-divergence, information distance and the dual $\alpha $-connection structure. The K\"ahler structure simplifies the calculation of the metric tensor, connection, Ricci tensor and scalar curvature, and the $\alpha $-generalization of the geometric objects. The Laplace--Beltrami operator is also simplified in the K\"ahler geometry. One of the goals in information geometry is the construction of Bayesian priors outperforming the Jeffreys prior, which we use to demonstrate the utility of the K\"ahler structure.